Photonic crystal and optical waveguide elements

ABSTRACT

A photonic crystal having a structure of which the refractive index changes with a periodicity defined in a polar coordinate system is disclosed. And an optical waveguide element comprising said photonic crystal, optical inlet and outlet regions on the surface of said photonic crystal, and a defect region of incomplete photonic crystal periodicity formed within said photonic crystal is disclosed. The defect region functions as an optical waveguide path by guiding from the inlet region to the outlet region an optical signal incapable of propagating through the photonic band gap of the photonic crystal.

TECHNICAL FIELD

[0001] The present invention relates to the technical areas of photoniccrystals and optical waveguide elements employing the same, and morespecifically, to the technical areas of photonic crystals capable offorming full photonic band gaps and optical waveguide elements employingthe same.

RELATED ART

[0002] Conventionally, structures having changing periodic refractiveindexes, in other words periodic dielectric structures, are known toblock light of a specific wavelength based on their periodicity (forexample, see Applied Physical Letters, Vol. 64 (16), pp. 2,059-2,062,and Physical Review Letters, Vol. 67 (17), pp. 2,295-2,298). This is aphenomenon based on the structure forming a photonic band gap similar tothe way in which normal crystals form electron band gaps. Thus, thestructure is referred to as a “photonic crystal.” A variety of researchhas been actively pursued in recent years. Photonic crystals arenormally produced by artificially imparting periodic change to therefractive index of a structure. For example, they can be produced byperiodically arranging areas of differing refractive indexes into astructure comprised of materials having specific refractive indexes.Conventional photonic crystals, mimicking crystals present in thenatural world, for example, exploit the periodicity based ontranslational symmetry that is observed in trigonal, tetragonal, andsimilar lattices.

[0003] When employing photonic crystals in various precision opticalequipment and the like, the photonic crystals are required to completelyblock light of specific wavelengths. However, conventional photoniccrystals having periodic structures of translational symmetry haveproblems in that they are dependent on the direction of propagation oflight, and the wavelengths of the photonic band gaps formed vary(referred to below as “directional dependence on photonic band gap”).Thus, in conventional photonic crystals, a broad range of photonic bandgaps are formed to impart overlapping gaps and ensure blocking of lightpropagating in all directions. Imparting a broad range of photonic bandgaps requires employing a structure of materials in which the photoniccrystals have large differences in refractive index. In the naturalworld, air has the lowest refractive index. However, forming regions ofair (holes and the like) in a structure compromises the strength of thestructure, limiting its applications and possibly creating designproblems in application to precision optical equipment. Further, anarrow range of materials from which to make selections, limitedmanufacturing methods, and the like are undesirable in practical terms.When it is possible to artificially impart changes in refractive indexcapable of forming a full photonic band gap in a photonic crystal, aphotonic crystal capable of blocking electromagnetic radiationpropagating in all directions is obtained. Such a photonic crystalaffords the advantages of markedly improved performance in opticaldevices employing the crystal, production advantages, and a broaderrange of possible applications.

[0004] In recent years, optical waveguide elements employing photoniccrystals have been the focus of great attention. These are devices inwhich linear defects of noncrystalline structure are formed in photoniccrystals, light is confined to the defects, and the light propagatesalong the defects. Since the optical waveguide paths of photoniccrystals have the property of strongly confining light by means of theexistence of photonic band gaps, they afford the advantage of less lightloss than conventional optical waveguide paths. Their application tovarious optical circuits is anticipated. However, two-dimensionalphotonic crystals having conventional trigonal and tetragonal latticesare problematic in that the introduction of defects is limited. Forexample, when forming optical waveguide paths with trigonal latticecrystals, linear defects can only be made to intersect at 60 and 120degrees, and when employing tetragonal lattice crystals, linear defectscan only be made to intersect at 90 degrees. Thus, in conventionalphotonic crystal optical waveguide paths, the angle of curvature in thedirection of light propagation ends up being limited. Were it possibleto freely bend the path of light being guided by a photonic crystaloptical waveguide, it would be possible to broaden the degree of freedomof design in optical circuit applications, facilitating design.

SUMMARY OF THE INVENTION

[0005] The present invention, devised in light of the variousabove-described problems, has for its object to provide a novel photoniccrystal capable of blocking light in all propagation directions andpermitting the formation of a full photonic band gap. The presentinvention also has the object of providing a novel photonic crystalwithout directional dependence of the photonic band gap. The presentinvention has the still further object of providing a novel photoniccrystal reducing restrictions on the selection of constituent materialsand on manufacturing. The present invention has the additional object ofproviding optical waveguide elements affording low optical loss and areduction in the restrictions on shape in the formation of opticalwaveguides. And the present invention has the still further object ofproviding an optical waveguide capable of reducing restrictions on thedesign of optical circuits when applied to optical circuits.

[0006] According to the present invention there is disclosed a photoniccrystal having a structure of which the refractive index changes with aperiodicity defined in a polar coordinate system.

[0007] There are also disclosed the photonic crystal in which refractiveindex changes between two values of n₁ and n₂ (where n₁ is not equal ton₂) based on a periodicity defined in a polar coordinate system; thephotonic crystal wherein said periodicity is of a non-translationalsymmetry; the photonic crystal wherein said periodicity has a rotationalsymmetry; the photonic crystal wherein the refractive index changesbased on a periodicity that is two-dimensionally defined in a polarcoordinate system; and the photonic crystal in which a structural unitcomprising a first material having a refractive index of n₁ and a secondmaterial having a refractive index of n₂ (where n₁ is not equal to n₂)repeatedly occurs at positions rotated by a θ degree (0<θ<<360) about apoint serving as the center of a polar coordinate system; the photoniccrystal in which a region having a refractive index differing from airin space is arranged with a periodicity defined by a polar coordinatesystem.

[0008] According to another aspect of the present invention there isdisclosed a photonic crystal comprising plural elements with arefractive index n₁, arranged in a two-dimensional rotational symmetrylattice; and plural spaces with a refractive index n₂ (where n₁ is notequal to n₂) between adjacent said elements.

[0009] According to further aspect of the present invention there isdisclosed an optical waveguide element comprising a photonic crystalhaving a structure of which the refractive index changes with aperiodicity defined in a polar coordinate system, optical inlet andoutlet regions on the surface of said photonic crystal, and a defectregion of incomplete photonic crystal periodicity formed within saidphotonic crystal, wherein said defect region functions as an opticalwaveguide path by guiding from said inlet region to said outlet regionan optical signal incapable of propagating through the photonic band gapof said photonic crystal.

[0010] There are also disclosed the optical waveguide element whereinsaid photonic crystal has the structure in which refractive indexchanges between two values of n₁ and n₂ (where n₁ is not equal to n₂)based on a periodicity defined in a polar coordinate system; the opticalwaveguide element wherein said photonic crystal periodicity is of anon-translational symmetry; the optical waveguide element wherein saidphotonic crystal periodicity has a rotational symmetry; the opticalwaveguide element wherein said photonic crystal periodicity istwo-dimensionally defined in a polar coordinate system; the opticalwaveguide element wherein said photonic crystal is in which a structuralunit comprising a first material with a refractive index of n₁ and asecond material with a refractive index of n₂ (where n₁ is not equal ton₂) repeatedly occurs at positions rotated by a θ degree (0<θ<<360)about a point serving as the center of a polar coordinate system; theoptical waveguide element wherein said photonic crystal is in which aregion having a refractive index differing from air in space is arrangedwith a periodicity defined by a polar coordinate system; the opticalwaveguide element wherein said defect region comprises at least one bendin the direction of light propagation; and the optical waveguide elementwherein said defect region comprises at least one region lying in acircular arc in the direction of light propagation.

[0011] According to another aspect of the present invention there isdisclosed an optical waveguide element comprising:

[0012] photonic crystal comprising plural elements with a refractiveindex n₁ arranged in a two-dimensional rotational symmetry lattice andplural spaces with a refractive index n₂ (where n₁ is not equal to n₂)between adjacent said elements;

[0013] optical inlet and outlet regions on the surface of said photoniccrystal; and

[0014] a defect region of incomplete said two-dimensional rotationalsymmetry lattice formed within said photonic crystal, wherein saiddefect region functions as an optical waveguide path by guiding fromsaid inlet region to said outlet region an optical signal incapable ofpropagating through the photonic band gap of said photonic crystal.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015]FIG. 1 shows examples of arrangements of the symmetries defined bya polar coordinate system permitting the application of the presentinvention.

[0016]FIG. 2 shows an example of an arrangement of the symmetry definedby a polar coordinate system permitting the application of the presentinvention.

[0017]FIG. 3 is a schematic showing a model of a photonic crystalrelating to the present invention.

[0018]FIG. 4 is a graph showing the results of calculation of theintensity of incident light in a model of the photonic crystal of thepresent invention.

[0019]FIG. 5 consists of plots of contour lines showing the results ofthe calculation of the absolute values of the pointing vector atspecific positions of incident light in a model of the photonic crystalof the present invention.

[0020]FIG. 6 consists of plots of contour lines showing the results ofthe calculation of the absolute values of the pointing vector atspecific positions of incident light in a model of the photonic crystalof the present invention.

[0021]FIG. 7 is a schematic of the symmetry of an example of a photoniccrystal relating to the present invention.

[0022]FIG. 8 is a schematic of examples of defect regions and symmetryin an optical waveguide element relating to the present invention.

[0023]FIG. 9 is a schematic of further examples of defect regions andsymmetry in an optical waveguide element relating to the presentinvention.

[0024]FIG. 10 is a schematic of further examples of defect regions andsymmetry in an optical waveguide element relating to the presentinvention.

[0025]FIG. 11 is a schematic of further examples of defect regions andsymmetry in an optical waveguide element relating to the presentinvention.

[0026]FIG. 12 is a schematic of further examples of defect regions andsymmetry in an optical waveguide element relating to the presentinvention.

[0027]FIG. 13 consists of graphs showing the results of calculating theelectric field intensities of models of photonic crystals relating tothe present invention.

[0028]FIG. 14 consists of graphs showing the results of computing theelectric field intensities of models of optical waveguide elementsrelating to the present invention.

[0029]FIG. 15 consists of graphs showing the results of computing thedistribution of intensities within the crystal in a model of an opticalwaveguide element relating to the present invention.

[0030]FIG. 16 consists of graphs showing the results of computing theelectric field intensities in a further model of an optical waveguideelement relating to the present invention.

[0031]FIG. 17 consists of graphs showing the results of computing thedistribution of intensities within the crystal of a further model of anoptical waveguide element relating to the present invention.

[0032]FIG. 18 is a photograph of a photonic crystal produced in anembodiment.

[0033]FIG. 19 is a schematic of the method of measuring the electricfield intensity within the photonic crystal in the embodiment.

[0034]FIG. 20 shows the spectrum of electric field intensity actuallymeasured for the photonic crystal produced in the embodiment.

[0035]FIG. 21 consists of graphs showing the results of calculations ofthe electric field intensity of a computation model corresponding to thephotonic crystal produced in the embodiment.

[0036]FIG. 22 is a graph showing the directional dependency of the nickbank gap actually measured for the photonic crystal produced in theembodiment.

DETAILED DESCRIPTION OF THE INVENTION

[0037] In contrast to conventional photonic crystals having periodicitydefined in a rectangular coordinate system, that is, having changes inrefractive index within the structure based on translational symmetry,the photonic crystal of the present invention is characterized by havingchanges in refractive index within the structure based on periodicitydefined in a polar coordinate system. This periodicity defined in apolar coordinate system may be either two-dimensional orthree-dimensional. In the present Specification, the term “photoniccrystal” does not mean a conventional photonic crystal having within itsstructure a change in refractive index having translational symmetry,but rather is used in a manner including all structures having internalchanges in refractive index capable of forming a photonic band gap. Solong as the requirements of the present invention are satisfied,photonic crystals not included in the conventional definition of“photonic crystals” fall within the scope of the present invention.

[0038] The phrase “periodicity defined in a polar coordinate system”means a state in which an identical base structure is arranged inrepeating fashion in polar coordinate space. Taking the example of atwo-dimensional polar coordinate system, this refers to a state having astructure repeatedly exhibiting the same arrangement at a positionrotated by a certain angle about a center point. There are variousperiodicities; examples are a periodicity having a two-rotation symmetryin which the same structural unit repeatedly appears with each rotationof 180°; a periodicity having a three-rotation symmetry in which thesame structural unit repeatedly appears with each rotation of 120°; anda periodicity having a four-rotation symmetry in which the samestructural unit repeatedly appears with each rotation of 90°. Further, aperiodicity having a single symmetry in which the same structural unitdoes not repeat unless rotated 360° is also included herein. FIGS. 1(a)through (d) show a three-rotation symmetry, four-rotation symmetry,five-rotation symmetry, and six-rotation symmetry, respectively. FIG. 2shows a single-rotation symmetry arrangement. The present invention canbe applied to all of these.

[0039] In a periodicity defined by a polar coordinate system, there arenumerous rotational symmetries. A periodicity having rotational symmetryis desirably employed in the present invention. Although periodicitiesdefined in polar coordinate systems may have arrangements that may beperiodic in rectangular coordinate systems, that is, may havetranslational symmetry, so long as there is a structure the refractiveindex of which changes based on a periodicity defined in a polarcoordinate system, irrespective of whether or not a translationalsymmetry exists, the structure is covered by the present invention.However, to reduce the directional dependency of the photonic band gap,a periodicity without translational symmetry is desirable.

[0040] The photonic crystal of the present invention is characterized byhaving a structure in which the refractive index changes based on aperiodicity defined by a polar coordinate system. Normally, in aphotonic crystal, there are two values defining the change in refractiveindex, and the greater the difference in the two values of refractiveindex, the greater the photonic band gap. The refractive index isspecific to the material. Examples of materials having high refractiveindexes of greater than or equal to 2 are: diamond (2.41), Si (about 3),TiO₂ (2.6), and Ta₂O₅ (2.3). Examples of materials having low refractiveindexes are SiO₂ and optical glass. As set forth above, in conventionalphotonic crystals, to reduce the direction dependency of the photonicband gap, it is necessary to maintain an extremely large photonicband-gap width. Thus, many structures are configured of a region oflower refractive index in the form of air (for example, holes or voids),and a region comprised of a material of high refractive index. Thisresults in drawbacks such as low strength and limited applications, aswell as production drawbacks such as limits to the materials that can beemployed. The photonic crystal of the present invention is configured ofTa₂O₅ or the like as a high refractive index range, and something otherthan air as the low refractive index range, such as a low refractiveindex material such as SiO₂, permitting the formation of a full photonicband gap. Nor does the photonic crystal of the present inventionpreclude the formation of air regions.

[0041] In the photonic crystal of the present invention, the period ofthe refractive index is not specifically limited, and may be determinedbased on the wavelength of the light that is to be blocked. For example,when the objective is to block visible light or near infrared radiation,it suffices to change the period of the refractive index by about thesame amount as the wavelength of visible light or near infraredradiation.

[0042] In the photonic crystal of the present invention, it suffices toincorporate into some portion thereof a structure in which therefractive index changes based on a periodicity defined by polarcoordinates. For example, photonic crystals configured by arrangingstructures in which the refractive index changes based on a periodicitydefined by polar coordinates into a two-dimensional arrangement havingtranslational symmetry are also covered by the photonic crystal of thepresent invention. As stated above, photonic crystals having a structurein which the refractive index changes based on a two-dimensionalperiodicity defined by polar coordinates are covered by the presentinvention; that is, photonic crystals having a structure having arefractive index that does not change based on a periodicity that isthree-dimensionally defined by polar coordinates are also covered by thepresent invention.

[0043] One embodiment of the present invention is a photonic crystalarranged with a two-dimensional periodicity defined in a polarcoordinate system by multiple rods comprised of a material having arefractive index n₁ (n₁>1, preferably n₁=1.6) in space (refractiveindex=1). The arrangement of two-dimensional periodicity can beaccomplished, for example, as the arrangement of rotational symmetryshown in FIG. 3. In FIG. 3, an “O” denotes the position of a rod, andthe number within the “O” is used to denote the position. In thearrangement shown in FIG. 3, the same array repeatedly appears atpositions rotated by 72° about the center (position 76) of the polarcoordinates, that is, is an arrangement having a five-rotation symmetry.It also has periodicity in a radial direction, and a periodicity inwhich concentric circles of rods arranged on the circumference of thecircle repeat with a single period. As will be understood from FIG. 3,the rod arrangement does not have periodicity in rectangularcoordinates; it is an arrangement without translational symmetry.

[0044] The following calculations were made for a photonic crystalhaving the structure with a refractive index changing based on theperiodicity of FIG. 3 to confirm directional independence of thephotonic band gap.

[0045] A structure obtained by arranging rods consisting of a materialwith a refractive index of 1.92 at the positions indicated by the 76circles shown in FIG. 3 with each pair of rods having a center distanceD (denoted by D in FIG. 3) between them of 85 micrometers was employedas the computational model. The diameter R₀ of the bottom surface of therods (denoted by R₀ in FIG. 3) was made 22 micrometers, and the heightof the rods was made infinitely large. The intensity of light reachingpositions a-e in the model when light corresponding to millimeter-waves(frequencies of from 0 to 200 cm⁻¹) was radiated in the direction of thearrow (A=0) was calculated. Although the photonic band gap present canbe demonstrated by irradiating light from within a photonic crystal andshowing that light of specific wavelength does not leak to the outside,due to the properties of light, the same demonstration can be effectedby reversing the arrival position and irradiation position of the light.

[0046]FIG. 4 gives the computation results. FIG. 4 shows graphs in whichthe frequency of light is plotted on the x-axis and the intensity of thelight on the y-axis for light arriving at positions a-e. Thecomputational results of FIG. 4 show that the light reaching any of thepositions had an intensity at a frequency of about 50 cm⁻¹ that wasextremely low. That is, in the model of the photonic crystal having aperiodicity defined in the polar coordinate system shown in FIG. 3, thecomputations demonstrate that a photonic band gap was formed for lighthaving a frequency of about 50 cm⁻¹. In these calculations, TM modepolarization was employed. However, such a photonic band gap can also beformed for TE mode polarization.

[0047] Next, to examine the directional dependence of the photonic bandgap, as shown in FIG. 3, the direction of incidence of the light wasdisplaced one degree at a time in direction θ (theta) over a total of 36degrees (36 points), and when light entered at the various angles, theintensity of the light reaching point c was calculated. Thecomputational results are given in FIGS. 5(a) and (b), with theintensity of light being denoted as contour lines. FIG. 5(a) shows theabsolute value of the pointing vector of light reaching position c forTE mode polarization and FIG. 5(b) shows the same for TM modepolarization. The x axis denotes the frequency cm⁻¹ (1 to 200, 400points), and the y axis denotes the displacement (angle of incidence) inthe direction of incidence of the light. The intensity of the absolutevalue of the pointing vector of the light is denoted by color darknessin the figures (by color in original figures). As will be understoodfrom the scale recorded beneath the graphs, the pointing vector was mostweak in the portions of dark color (dark blue in original figures). Inboth FIGS. 5(a) and (b), there were regions of low pointing vectors neara frequency of 50 cm⁻¹ that are denoted as areas of dark color bands inthe contour line drawings. These results indicate that in this photoniccrystal model, the wavelength of the photonic band gap did not vary evenwhen the direction of incident light was varied.

[0048] In the contour line drawings, analytic computations were made bythe two-dimensional vector cylindrical-function expansion method. Thiswas also the case in FIG. 6 below. Although the angle of incidence ofthe light was not rotated through 72 degrees (to the point ofequivalence in a five-rotation symmetry), it may be presumed from thesymmetry of the arrangement that similar computational results would beobtained in a 36-degree rotation.

[0049] Similarly, the absolute values of the pointing vectors of thelight reaching the various positions were calculated when the degree ofincidence of the light was varied over a total of 36 degrees (36 points)for the positions of rods 4, 5, 9, 10, 14, and 16 in FIG. 3. The resultsare given in FIG. 6. FIG. 6 consists of contour graphs of the absolutevalues of the pointing vectors of light reaching the positions of rods4, 5, 9, 10, 14, and 15 (calculated for TM mode polarization). Thenumber recorded at the upper left of each of the contour line graphsindicates the position of the rod. In the contour line graphs of all ofthe positions of FIG. 6, there was no dependence on direction, there wasa region where the pointing vector was low near a frequency of 50 cm⁻¹,and there was a band-shaped region of concentrated color (dark blue inoriginal figures). These results indicate that even when the directionof entry of the light was varied in the model of this photonic crystal,there was almost no change in the wavelength of the photonic band gapnot only at the center point (position 76), but also at positions to theoutside. That is, this shows that there was no position and incidentangle dependence of the photonic band gap.

[0050] It is assumed from the periodicity that calculations of pointingvectors for positions equivalent to positions 4, 5, 9, 10, 14, and 15would yield similar results.

[0051] From these computational results it will be understood that whena material of relatively low refractive index (1.92) is employed, thatis, when the difference in refractive index in the photonic crystal waslow, even in an embodiment where it is not possible to form a widephotonic band gap, the photonic crystal of the present invention wasable to block the light in all directions of propagation by forming afull photonic band gap. As is apparent in the results, a model in whichthe rods were comprised of a material with a comparatively lowrefractive index was employed in the above calculations. However, thesame results would be achieved were the columns to be configured of amaterial of high refractive index. Further, in the above-describedmodel, the computations were performed for electromagnetic waves on theorder of millimeter-waves, but similar results would be obtained forcomputations on visible light to near infrared radiation. That is, withthe photonic crystal of the embodiment, it is possible to form photoniccrystals for visible light to near infrared radiation. In thisembodiment, it suffices to position the rods to achieve a period with awavelength of from visible light to near infrared.

[0052] In the above calculations, an arrangement with five-rotationsymmetry was employed. However, it is possible to configure photoniccrystals employing arrays having other than five-rotational symmetries.Further, in the above calculations, a photonic crystal having astructure in which rods having a round bottom surface were arranged.However, it is presumed that similar results would be achieved withphotonic crystals in which polygonal rods having triangular,rectangular, and hexagonal bottom surfaces were arranged.

[0053] In the above-described embodiment, a photonic crystal having aconfiguration in which rods of a material other than air were arrangedin air. However, it is also possible to manufacture the photonic crystalof the present invention by machining holes in a structure of relativelyhigh refractive index other than air, and forming multiple rods of airor some other material (a material of relatively low refractive index,such as SiO₂) in the structure. Semiconductors such as Si and GaAs arematerials of comparative high refractive index for which micromachiningtechnology is highly developed. These materials are thus desirable foruse as the above-mentioned structures. For example, Sugimoto, Y. et al.,Journal of Applied Physics, Vol. 91, p. 922 (2002), FIG. 4 and 5,disclose examples of the machining of trigonal lattice holes insemiconductors. The photonic crystal of the present invention can befabricated by machining holes at the positions denoted by “O” in thefigures based on the polar coordinates given in FIGS. 1 through 3 on thex-y plane of a structure comprised of a semiconductor using thetechniques employed in the cited literature. The use of micromachiningtechniques permits the fabrication of photonic crystals in which aphotonic gap band is present for light in the visible and infraredregions.

[0054] In a further embodiment of the present invention, a photoniccrystal of a structure in which a structural unit comprised of amaterial (other than air) having a refractive index n₁ (other than air)and a material having a refractive index n₂ (where n₁ is not equal ton₂) is arranged with a periodicity defined by polar coordinates,specifically, in which the constitutional unit of the second material(other than air) has a structure repeating at a rotational position ofprecisely a certain angle about a center in the form of some point inthe coordinate axis system. The present embodiment affords the advantageof high strength, since no space is formed in the structure. As is clearfrom the computational results set forth above, the incorporation of astructure having a refractive index changing with a periodicity definedby polar coordinates reduces the wavelength shift of the photonic bandgap. Accordingly, due to the large difference in refractive index withinthe structure, there is no need to form voids, and, for example, even ina configuration comprising a combination of materials of comparativelylow difference in refractive index, such as SiO₂ and Ta₂O₅, it ispossible to block light in all directions of propagation.

[0055] The photonic crystal of the present implementation mode may beproduced, for example, by a manufacturing method comprising a step offorming on a substrate surface irregularities arranged with atwo-dimensional periodicity defined by a polar coordinate system, and astep of stacking a layer comprised of a material having a refractiveindex of n₁ and a layer having a refractive index of n₂ on the substratesurface having irregularities. When targeting a photonic crystal forvisible to near infrared radiation, the irregularities formed on thesubstrate surface must be minute structures. In this case, it isdesirable to form the irregularities by electron-beam lithography. Anembodiment where the refractive index changes with three-dimensionalperiodicity, a structure in which the refractive index changes withthree-dimensional periodicity can be manufactured by, for example,stacking multiple films (for example, films comprised of a materialhaving a refractive index of n₁ and a refractive index of n₂) on asubstrate having surface irregularities by bias sputtering or the like.

[0056] The photonic crystal of the present invention can be used tocontrol electromagnetic waves, and is desirably employed as an opticalwaveguide element to control the direction of propagation of light.Further, an optical waveguide element employing the photonic crystal ofthe present invention can be employed in three-dimensional opticalcircuits, optical devices, and light-emitting devices. Since thephotonic crystal of the present invention forms a full photonic bandgap, when employed in light-emitting devices, for example, it can beexpected to substantially improve light-emission efficiency.

[0057] An embodiment applying the photonic crystal of the presentinvention as a light waveguide element will be described next.

[0058] This embodiment of the present invention is an optical waveguideelement comprised of the photonic crystal of the present invention,inlet and outlet regions formed in the surface of the photonic crystal,and a defect region formed in the photonic crystal and impartingincomplete periodicity to the photonic crystal. The defect regionfunctions as an optical waveguide path guiding from the inlet region tothe outlet region light that is incapable of propagating due to thephotonic band gap of the photonic crystal.

[0059] In the optical waveguide element of the present invention, thepresence of the photonic band gap of the photonic crystal confinesentering light to the defect region formed in the photonic crystal andacts as an optical waveguide permitting the propagation of the confinedlight along the defect region. In the optical waveguide element of thepresent invention, the use of the presence of the photonic band gap ofthe photonic crystal to confine light results in little loss of light byreflection or the like. Further, in the present invention, the use of aphotonic crystal in which the index of refraction changes by having aperiodicity defined by a polar coordinate system, the restrictions onthe shape of the defect region that can be internally formed arereduced, and defect regions of various shapes can be formed.Accordingly, it is possible to readily form defect regions causing lightto propagate in a desired direction, and when applied to opticalcircuits, permits a widening of the degree of freedom in the designingof optical circuits.

[0060] This embodiment of the optical waveguide element of the presentinvention is an optical waveguide element employing a photonic crystalhaving a two-dimensional periodicity of rotational symmetry in which astructural unit comprised of a first material of refractive index n₁ anda second material of refractive index n₂ (where n₁ does not equal n₂)occurs repeatedly at positions of prescribed rotation about some pointas center of a polar coordinate system. An optical waveguide elementemploying a photonic crystal having a periodicity with rotationalsymmetry in the x-y plane (letting the z direction be infinitely large)shown in FIG. 7 will be described as an embodiment of the presentinvention. A photonic band gap based on a periodic structure offive-rotation symmetry is present in the x-y plane in a structure inwhich are arranged multiple rods (letting the z direction be infinitelylarge) comprised of a material with a refractive index of n₂ (n₂ beingdifferent from 1) at the positions denoted by “O” in FIG. 7 on the x-yplane (refractive index n₁=1) in (x, y, z) space.

[0061] An optical waveguide element can be obtained by forming a defectregion of incomplete periodicity in the photonic crystal exhibitingrotational symmetry shown in FIG. 7. One example of optical waveguideelement of the present invention can be fabricated by taking two pointson the circumference as the light inlet and outlet regions and removingthe rods from a region connected through the center of the circlepassing through these two points to form a defect region (for example,the space in which no “O”s are arranged in FIG. 8, with the z directionbeing infinite space), which functions as an optical waveguide path.Another example can be fabricated by taking two points on thecircumference as the light inlet and outlet regions and removing thesome rods from a region connected through the center of the circlepassing through these two points to form a defect region (for example,the space in which no “O”s are arranged at interval of one “O” in FIG.12, with the z direction being infinite space), which functions as anoptical coupled cavity waveguide path. In a conventional opticalwaveguide element employing a photonic crystal having a trigonal ortetragonal lattice of two-dimensional periodicity, it is only possibleto incorporate a linear defect intersecting at 60 degrees, 120 degrees,or 90 degrees due to the periodicity of the photonic crystal. However,in the optical waveguide element of the present invention, the range ofselection of the angle of bend of the defect region is broadened. Inphotonic crystals exhibiting two-dimensional rotational symmetry, it ispossible to take two points on the circumference as the optical inletand outlet, form a defect region linking these two points through thecenter of the circle, and bend the direction of light propagation by anyangle based on the number n of rotational symmetry.

[0062]FIGS. 8 through 10 show embodiments in which points on thecircumference of the photonic crystal shown in FIG. 7 are taken as theoptical inlet and outlet and defect regions are formed along axes ofrotational symmetry.

[0063] The optical waveguide element shown in FIG. 8 is an embodiment inwhich the points of intersection of two of the five-rotation symmetryaxes with the circumference are taken as an optical inlet SL0 and outletSL2, and a defect region is formed linking point SL0 and point SL2through the center SL1 of the circle. Light enters through point SL0, isconfined to the defect region, propagates along the defect region, andarrives at point SL2. The optical waveguide element of the periodicstructure shown in FIG. 8 permits the bending of the direction ofpropagation by about 144 degrees while controlling the loss of light.Further, by varying the combination of the five-rotation symmetry axesforming the defect region, it is possible to configure an opticalwaveguide element bending by about 72 degrees in the direction ofpropagation of the light.

[0064] The optical waveguide element shown in FIG. 9 is an embodiment inwhich the two points SL2 and SL4 on the circumference are taken asoutlets, point SL0 is taken as inlet, and a defect region is formedlinking these points through the center SL1 of the circle. In thisoptical waveguide element, there is a fork in the direction of lightpropagation. In FIG. 9, when light entering through SL0 propagates alongthe defect region and reaches SL1, it branches about 36 degrees eachboth right and left, or a total of 72 degrees, reaching points SL2 andSL4. The optical waveguide element of FIG. 10 is an embodiment in whichrods positioned along all five of the five-rotation axes are removed toform defect regions. In this optical waveguide element, when lightentering at point SL0 propagates along the defect regions and reachesthe center SL1 of the circle, it branches in four directions along thedefect regions, reaching SL2 through SL5.

[0065] Further, in a photonic crystal exhibiting two-dimensionalrotational symmetry as shown in FIG. 11, a photonic crystal in the shapeof a fan defined by two radii comprised of straight lines and a segmentof the circumference in the form of a curved line can be used to form anoptical waveguide element. Not only can this photonic crystal be used asthe above-described optical waveguide element having a bend andbranching portions, it can also be used as an optical waveguide elementin which the direction of advance of the light is an arc-shaped bend.For examples, as shown in FIG. 11, light inlet SL0 and outlet SL2 can beformed in radii and a defect connecting SL0 and SL2 in an arc can beformed to configure an optical waveguide element in which the directionof advancement of light is bent in an arc.

[0066] There are shown embodiments of optical defect waveguide elementsin FIGS. 8 through 11, however, the photonic crystal of the presentinvention may also be employed coupled cavity waveguide. The opticalwaveguide element shown in FIG. 12 is a coupled cavity waveguide elementwhich may be fabricated by removing rods at interval of a rod, notremoving as all rods as FIG. 8, from a region connected through thecenter passing through SL0 to SL2 to form a defect region in thephotonic crystal as shown in FIG. 7.

[0067] The defect region shown in FIG. 8 was formed in the photoniccrystal shown in FIG. 7 and the following calculations were made of thetransmitted spectrum at position SL2 in the figure to confirm operationas an optical waveguide element.

[0068] A structure in which the rods made of a material having arefractive index of 1.92 (dielectric constant of 3.7) were arranged atthe positions denoted by circles “O” in FIG. 7, with a center distanceof D=85 micrometers between pairs of rods on the five-rotationalsymmetry axes, was adopted as the computation model and the internalelectric field intensities were calculated. The radius of the bottomsurface of the rods was made 22 micrometers and the height of the rodswas assumed to be infinite. The results are given in FIG. 13. FIG. 13consists of graphs showing the results with light frequency plotted onthe x-axis and light intensity on the y-axis. The computational resultsof FIG. 13 reveal that a photonic band gap was present in the structureof FIG. 7 with its center at 50 cm⁻¹.

[0069] Next, the transmitted spectrum at SL2 was computed from acomputational model obtained by removing multiple rods from thestructure shown in FIG. 7 to form an internal defect region and achievethe structure shown in FIG. 8. Specifically, the rods positioned on twoof the five-rotation symmetry axes, or a total of 19 rods, were removedfrom positions of five-rotation symmetry in FIG. 7 to form defectregions. It was assumed that light corresponding to millimeter-waves(frequency of from 0 to 200 cm⁻¹) was irradiated at SL0, and thetransmitted spectrum reaching position SL2 was computed.

[0070]FIG. 14 gives the computational results. FIG. 14 is a graphshowing the computation results with light frequency plotted on thex-axis and transmission efficiency plotted on the y-axis. In FIG. 14, atransmission efficiency peak is present near 52 cm⁻¹, the point wherethe light intensity was 0 in FIG. 13. The results reveal that atransmission mode was present in the structure of FIG. 8. As shown inFIG. 13, a photonic band gap was present in the structure of FIG. 7,with 40 to 80 cm⁻¹ electromagnetic waves being unable to pass throughthe structure. However, in the model in which the defect region shown inFIG. 8 was formed, propagation of electromagnetic waves was possible inthe defect range in the vicinity of 52 cm⁻¹. That is, the fact that thestructure shown in FIG. 8 functioned as an optical waveguide element wasproven by the computations.

[0071]FIG. 15 shows the results of computations of the intensitydistribution within a crystal with a transmission mode ofelectromagnetic waves in the vicinity of 52 cm⁻¹. The intensitydistribution is indicated by color density (actual color) in the figure,with dark colored portions (actual red) indicating the greatestintensity on the scale indicated beneath the graph. The computationalresults of the intensity distribution shown in FIG. 15 indicate thatelectromagnetic radiation near 52 cm⁻¹ propagated along the opticalwaveguide path in the form of the defect region.

[0072] Next the transmitted spectrum at SL2 was computed taking thestructure shown in FIG. 9, in which multiple columns were removed fromthe structure shown in FIG. 7 to form a branching defect region, as thecomputational model. Specifically, a total of 28 rods were removed frompositions on three of the five rotation symmetry axes in thefive-rotation symmetry arrangement shown in FIG. 7 to form defectregions. It was assumed that SL0 was irradiated with light correspondingto millimeter-waves (frequencies of from 0 to 200 cm⁻¹), and spectrum oftransmitted light reaching position SL2 was calculated. The results aregiven in FIG. 16. The results of the calculation of the intensitydistribution within a crystal of that transmission mode are given inFIG. 17.

[0073] As in the computation results shown in FIG. 14, a transmissionmode was present in the vicinity of 52 cm⁻¹ in FIG. 16. FIG. 17, showingthe intensity distribution within a crystal of that transmission mode,indicates that electromagnetic radiation entering at SL0 branchedevenly, reaching SL2 and SL4.

[0074] For the sake of simplicity, the above-described embodiments havedescribed the use of the photonic crystal of the present invention indefect waveguides. However, the photonic crystal of the presentinvention may also be employed in coupled cavity waveguides.

EXAMPLES

[0075] The present invention is described in detail below throughembodiments. The materials, proportions, operations and the like givenin the embodiments below may be suitably varied without departing fromthe spirit of the present invention. Accordingly, the scope of thepresent invention is not limited to the specific examples given below.

[0076] The structure shown in FIG. 18 was fabricated. The structureshown in FIG. 18 was comprised of air (refractive index 1.0) andmultiple acrylic rods (refractive index 1.61) and had a two-dimensionalperiodicity with five-rotation symmetry in the x-y plane, and could beassumed to have a shape that was infinite in the z direction.Specifically, the structure was fabricated by arranging 700 acrylic rods300 mm in length and 3 mm in radius at (x, y) coordinates satisfying thefollowing equations:

x=R×sin {(360×n)/5N}

y=R×cos {(360×n)/5N}

[0077] In the equations, R denotes the radius, given by the spacing ofthe acrylic rods multiplied by N. The actual spacing of the acrylic rodswas 12 mm. N corresponds to the number of concentric circles, and is aninteger of from 0 to 20. n denotes the number of rods (from 1 to 5) on asingle radius (R).

[0078] In this structure, the internal electric field intensity wasmeasured. A schematic of the method employed is given in FIG. 19.Electromagnetic radiation generated by a Network Analyzer (HP8510C) wasemitted through a horn antenna and directed onto the structure through alens. The electric field intensity within the structure was detectedwith a probe antenna. FIG. 20 gives the spectrum of the electric fieldintensity detected with this test system. FIG. 20 is a graph showing themeasurement results in which the electric field intensity is plotted onthe y-axis and the frequency is plotted on the x-axis. In the graph, twosets of measurement results are superimposed. FIG. 20 reveals thatwithin this structure, the electric field intensity was reduced sharplyin a zone centered on 12 GHz. That is, the structure of FIG. 18 wasfound to be a photonic crystal with a photonic band gap present.

[0079] Since the results obtained by the above-described test systemwere found to match the calculated results, a computation model of thephotonic crystal shown in FIG. 18 was employed to calculate theintensity of light entering at various positions in the photonic crystalby the same method used to derive the computation results given in FIG.4. However, in the computation used to derive the results given in FIG.4, the refractive index of the rods was 1.92, the radius of the rods was11 micrometers, and the distance between rods was 85 micrometers.However, in the present computations, to achieve correspondence with theresults obtained with the above-described test system, the refractiveindex of the rods was made 1.61, the radius of the rods was made 3 mm,and the distance between rods was made 12 mm. Further, in the model ofFIG. 18, since the number of concentric circles in (x, y) coordinateswas more than in the computational model of FIG. 3, the number ofmeasurement positions was increased from 5 to 8. The results are givenin FIG. 21. These computational results also exhibited a marked decreasein electric field intensity centered at about 12 GHz.

[0080] The direction of incident light was varied and the dependence ondirection of the photonic band gap was actually measured for thestructure of FIG. 18. The results are given in FIG. 22. FIG. 22 is agraph showing that when the entry direction was changed from 0 to 90°,the intensity dropped to below −10 dB, that is, it is a graph showingthe frequency range in which the photonic band gap was present. As willbe clear from the test results of FIG. 22, the structure of FIG. 7exhibited nearly identical photonic band gap frequencies even when theangle of light incidence was varied. These test results conform to thecalculated results given in FIG. 5.

[0081] These test results reveal that a photonic band gap was present inthe structure of FIG. 18 and that there was no directional dependence ofthe photonic band gap. Further, these test results match the calculatedresults, suggesting that if computation proves the presence of aphotonic band gap and no dependence thereof on a direction in thephotonic crystal of the present invention, the same results areobtainable experimentally.

[0082] In the photonic crystal fabricated in the embodiment, a band gapfor light on the millimeter-wave level has been disclosed. Actuallyemployed, however, are photonic crystals exhibiting a photonic band gapfor light at wavelengths within the infrared to visible range. Photoniccrystals corresponding to light in the infrared to visible range can byfabricated by machining holes in the above-described semiconductor andforming multiple rods comprised of air in the semiconductor at multiplepositions having two-dimensional rotational symmetry. A photonic crystalthus fabricated can be presumed to exhibit the same good characteristicsas demonstrated by the present embodiment.

[0083] The present invention provides a novel photonic crystal in whichis formed a full photonic band gap capable of blocking light in alldirections of propagation. The present invention further provides anovel photonic crystal without directional dependence of the photonicband gap. Still further, the present invention provides a novel photoniccrystal permitting a reduction in restrictions on the selection andfabrication of constituent materials. Still further, the presentinvention provides an optical waveguide element affording little loss oflight and reducing the limits on the direction of bending of thepropagation path of the light. Still further, the present inventionprovides an optical waveguide element broadening the scope of freedom inthe design of optical circuits when applied to optical circuits.

[0084] Having described our invention as related to the presentembodiments, it is our intention that the invention not be limited byany of the details of the description, unless otherwise specified, butrather be construed broadly within its spirit and scope as set out inthe accompanying claims.

What is claimed is:
 1. A photonic crystal having a structure of whichthe refractive index changes with a periodicity defined in a polarcoordinate system.
 2. The photonic crystal of claim 1, in whichrefractive index changes between two values of n₁ and n₂ (where n₁ isnot equal to n₂) based on a periodicity defined in a polar coordinatesystem.
 3. The photonic crystal of claim 1, wherein said periodicity isof a non-translational symmetry.
 4. The photonic crystal of claim 1,wherein said periodicity has a rotational symmetry.
 5. The photoniccrystal of claim 1, wherein the refractive index changes based on aperiodicity that is two-dimensionally defined in a polar coordinatesystem.
 6. The photonic crystal of claim 1, in which a structural unitcomprising a first material having a refractive index of n₁ and a secondmaterial having a refractive index of n₂ (where n₁ is not equal to n₂)repeatedly occurs at positions rotated by a θ degree (0<θ<<360) about apoint serving as the center of a polar coordinate system.
 7. Thephotonic crystal of claim 1, in which a region having a refractive indexdiffering from air in space is arranged with a periodicity defined by apolar coordinate system.
 8. A photonic crystal comprising pluralelements with a refractive index n₁, arranged in a two-dimensionalrotational symmetry lattice; and plural spaces with a refractive indexn₂ (where n₁ is not equal to n₂) between adjacent said elements.
 9. Anoptical waveguide element comprising a photonic crystal having astructure of which the refractive index changes with a periodicitydefined in a polar coordinate system, optical inlet and outlet regionson the surface of said photonic crystal, and a defect region ofincomplete photonic crystal periodicity formed within said photoniccrystal, wherein said defect region functions as an optical waveguidepath by guiding from said inlet region to said outlet region an opticalsignal incapable of propagating through the photonic band gap of saidphotonic crystal.
 10. The optical waveguide element of claim 9, whereinsaid photonic crystal has the structure in which refractive indexchanges between two values of n₁ and n₂ (where n₁ is not equal to n₂)based on a periodicity defined in a polar coordinate system.
 11. Theoptical waveguide element of claim 9, wherein said photonic crystalperiodicity is of a non-translational symmetry.
 12. The opticalwaveguide element of claim 9, wherein said photonic crystal periodicityhas a rotational symmetry.
 13. The optical waveguide element of claim 9,wherein said photonic crystal periodicity is two-dimensionally definedin a polar coordinate system.
 14. The optical waveguide element of claim9, wherein said photonic crystal is in which a structural unitcomprising a first material with a refractive index of n₁ and a secondmaterial with a refractive index of n₂ (where n₁ is not equal to n₂)repeatedly occurs at positions rotated by a θ degree (0<θ<<360) about apoint serving as the center of a polar coordinate system.
 15. Theoptical waveguide element of claim 9, wherein said photonic crystal isin which a region having a refractive index differing from air in spaceis arranged with a periodicity defined by a polar coordinate system. 16.The optical waveguide element of claim 9, wherein said defect regioncomprises at least one bend in the direction of light propagation. 17.The optical waveguide element of claim 9, wherein said defect regioncomprises at least one region lying in a circular arc in the directionof light propagation.
 18. An optical waveguide element comprising:photonic crystal comprising plural elements with a refractive index n₁arranged in a two-dimensional rotational symmetry lattice and pluralspaces with a refractive index n₂ (where n₁ is not equal to n₂) betweenadjacent said elements; optical inlet and outlet regions on the surfaceof said photonic crystal; and a defect region of incomplete saidtwo-dimensional rotational symmetry lattice formed within said photoniccrystal, wherein said defect region functions as an optical waveguidepath by guiding from said inlet region to said outlet region an opticalsignal incapable of propagating through the photonic band gap of saidphotonic crystal.
 19. The optical waveguide element of 18, wherein saiddefect region comprises radially extending portion.
 20. The opticalwaveguide element of 18, wherein said defect region comprises circularlyextending portion.